STAT 52800 Introduction to Mathematical Statistics

Zhankun Luo

Last updated on Aug 22, 2022

Notes:

problems of inference
Exponential family
statistical models: distributions of Exponential family, heavy tails and kewness
decision theory: evaluating an estimator and risk function, maximum risk and minimaxity
Exponential families, moments and moment generating function
point estimations
- likelihood function, MLE, score function
- Fisher information, sufficient statistics and factorization theorem,
- minimal sufficient statistics
- ancillary statistics and MLE in curved Exponential families, Basu’s theorem
- Rao-Blackwell theorem, uniformly minimum variance unbiased estimate (UMVUE)
- completeness, Lehmann-Scheffe theorem,
- asymptotic unbiasedness
- Cramer-Rao inequality
asymptotic approximations
- consistency of estimator sequence
- Cramer-Rao Conditions for consistency of MLE
- asymptotic distribution of MLE, Fisher information matrix
- pivot, Wald confidence intervals
- Delta theorem of Cramer, variance stabilizing transformations (VST)
- Rao’s score intervals, Wald confidence intervals and Slutsky’s theorem
testing of hypotheses and confidence regions
- type I and type II error probabilities
- randomized and nonrandomized tests
- most powerful tests, Neyman-Pearson lemma

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